Particle acceleration

From Wikipedia, the free encyclopedia
This article is about small-scale particle acceleration in acoustics. For acceleration of charged particles to very high energies, see particle accelerator.

In a compressible sound transmission medium - mainly air - air particles get an accelerated motion: the particle acceleration or sound acceleration with the symbol a in metre/second². In acoustics or physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. It is thus a vector quantity with dimension length/time². In SI units, this is m/s².

To accelerate an object (air particle) is to change its velocity over a period. Acceleration is defined technically as "the rate of change of velocity of an object with respect to time" and is given by the equation

{\mathbf {a}}={d{\mathbf {v}} \over dt}

where

  • a is the acceleration vector
  • v is the velocity vector expressed in m/s
  • t is time expressed in seconds.

This equation gives a the units of m/(s·s), or m/s² (read as "metres per second per second", or "metres per second squared").

An alternative equation is:

{\mathbf {{\bar {a}}}}={{\mathbf {v}}-{\mathbf {u}} \over t}

where

{\mathbf {{\bar {a}}}} is the average acceleration (m/s²)

{\mathbf {u}} is the initial velocity (m/s)

{\mathbf {v}} is the final velocity (m/s)

t is the time interval (s)

Transverse acceleration (perpendicular to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have

{\mathbf {a}}=-{\frac {v^{2}}{r}}{\frac {{\mathbf {r}}}{r}}=-\omega ^{2}{\mathbf {r}}

One common unit of acceleration is g-force, one g being the acceleration caused by the gravity of Earth.

In classical mechanics, acceleration a\ is related to force F\ and mass m\ (assumed to be constant) by way of Newton's second law:

F=m\cdot a

Equations in terms of other measurements

The Particle acceleration of the air particles a in m/s² of a plain sound wave is:

a=\xi \cdot \omega ^{2}=v\cdot \omega ={\frac {p\cdot \omega }{Z}}=\omega {\sqrt {\frac {J}{Z}}}=\omega {\sqrt {\frac {E}{\rho }}}=\omega {\sqrt {\frac {P_{{ac}}}{Z\cdot A}}}
Symbol Units Meaning
a m/s² particle acceleration
v m/s particle velocity
ξ m, meters particle displacement
\omega = 2 · \pi · f radians/s angular frequency
f Hz, hertz frequency
p Pa, pascals sound pressure
Z N·s/m³ acoustic impedance
J W/m² sound intensity
E W·s/m³ sound energy density
Pac W, watts sound power or acoustic power
A m² area

See also

External links

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.